Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19883 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | #61436 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
877 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,6,3,10,9)(5,11,12,13,15,8,14,16)(17,18)(19,29)(20,30)(21,25)(22,27)(23,26)(24,28)(31,32);(1,3)(2,6)(4,9)(5,13)(7,10)(8,12)(11,14)(15,16)(17,19)(18,20)(21,28)(22,25)(23,30)(24,31)(26,32)(27,29);(1,4,6,10)(2,7,3,9)(5,12,15,14)(8,16,11,13)(17,20,24,21)(18,19,28,31)(22,29,23,32)(25,26,30,27);(1,5,6,15)(2,8,3,11)(4,12,10,14)(7,16,9,13)(17,21,24,20)(18,25,28,30)(19,26,31,27)(22,29,23,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,c^b=ci,d^b=de,e^b=eh,f^c=fi,e^d=eh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 128335);